High-Dimensional Menger-Type Curvatures - Part I: Geometric Multipoles and Multiscale Inequalities

We define a discrete Menger-type curvature of d+2 points in a real separable Hilbert space H by an appropriate scaling of the squared volume of the corresponding (d+1)-simplex. We then form a continuous curvature of an Ahlfors d-regular measure on H by integrating the discrete curvature according to the product measure. The aim of this work, continued in a subsequent paper, is to estimate multiscale least squares approximations of such measures by the Menger-type curvature. More formally, we show that the continuous d-dimensional Menger-type curvature is comparable to the ``Jones-type flatness''. The latter quantity adds up scaled errors of approximations of a measure by d-planes at different scales and locations, and is commonly used to characterize uniform rectifiability. We thus obtain a characterization of uniform rectifiability by using the Menger-type curvature. In the current paper (part I) we control the continuous Menger-type curvature of an Ahlfors d-regular measure by its Jones-type flatness.Comment: 47 pages, 13 figures. Minor revisions and the inclusion of figure

Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations $\omega_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ on $K$ (for `nice' weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as $N$ grows large. Furthermore, if $K$ is an $\alpha$-rectifiable compact subset of Euclidean space ($\alpha$ an integer) with positive and finite $\alpha$-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as $N\to \infty$) a prescribed positive continuous limit distribution with respect to $\alpha$-dimensional Hausdorff measure. As a consequence of our energy related results for the unweighted case, we deduce that if $A$ is a compact $C^1$ manifold without boundary, then there exists a sequence of $N$-point best-packing configurations on $A$ whose mesh-separation ratios have limit superior (as $N\to \infty$) at most 2

Loudly sing cuckoo : More-than-human seasonalities in Britain

This research was funded by a grant from the Arts and Humanities Research Council, grant number AH/E009573/1.Peer reviewedPostprin

Toward three-dimensional simulations of stellar core collapse with magnetic fields

In spherical symmetry, very reliable models of stellar core collapse, bounce, and the postbounce phase can be constructed based on general relativistic Boltzmann neutrino transport. However, even if the time-integrated neutrino luminosity in the region between the surface of the protoneutron star and the stalled accretion shock is one or two orders of magnitude larger than the energy of a supernova explosion, it is generally accepted that the net energy transfer is not efficient enough to drive an explosion, unless the fluid instabilities in this regime are taken into account. Complementary to other groups, who are elaborating an extension of the accurate neutrino physics to axisymmetric simulations, we construct efficient parameterizations of the neutrino physics that enable three-dimensional magneto-hydrodynamics simulations that do not constrain the fluid instabilities by artificially imposed symmetries. We evaluate our approximations with respect to spherically symmetric Boltzmann neutrino transport, present preliminary MHD simulations with a resolution of 600 zones cubed, and illustrate the questions that can be addressed by this approac

Radiative transfer and the energy equation in SPH simulations of star formation

We introduce and test a new and highly efficient method for treating the thermal and radiative effects influencing the energy equation in SPH simulations of star formation. The method uses the density, temperature and gravitational potential of each particle to estimate a mean optical depth, which then regulates the particle's heating and cooling. The method captures -- at minimal computational cost -- the effects of (i) the rotational and vibrational degrees of freedom of H2, H2 dissociation, H0 ionisation, (ii) opacity changes due to ice mantle melting, sublimation of dust, molecular lines, H-, bound-free and free-free processes and electron scattering; (iv) external irradiation; and (v) thermal inertia. The new algorithm reproduces the results of previous authors and/or known analytic solutions. The computational cost is comparable to a standard SPH simulation with a simple barotropic equation of state. The method is easy to implement, can be applied to both particle- and grid-based codes, and handles optical depths 0<tau<10^{11}.Comment: Submitted to A&A, recommended for publicatio

Experimental observations of tsunami induced scour at onshore structures

Tsunami inundation of the coastal environment can induce scour at structure foundations leading to failure. A series of experiments are made using a unique Pneumatic Long Wave Generator to generate tsunami wave periods of 25 - 147 s equating to 3 - 17.3 mins at 1:50 Froude scale. The waves propagate over a sloping bathymetry and impinge upon a square structure founded onshore in a flat sediment bed. Flow velocity, height and scour are recorded as a function of time during tsunami inundation. The rate of scour is observed to be time dependent. Equilibrium, which is not attained, is argued to be an inappropriate measure for time-dependent transient flows such as tsunami in which the flow velocity, depth and direction are variable. The maximum scour depth is recorded and critically is observed not to be equal to the final depth due to significant sediment slumping when flow velocities reduce in the latter stages of inundation. Current and wave scour predictor equations over predict the scour, while the ASCE 7-16 method under predicts. Comparisons with available data in the literature show longer inundation durations increase the amount of scour

Heavy Isotope Composition of Oxygen in Zircon from Soil Sample 14163: Lunar Perspective of an Early Ocean on the Earth

Thirty oxygen analyses of a large (sub-millimetre) zircon grain from the lunar soil sample 14163 have been determined using CAMECA 1270 ion microprobe. The sample 14163 was returned form the Fra Mauro region by Apollo 14 mission. Zircon grain of 0.6-0.8 mm in size extracted from the sample was imaged using CL detector fitted to the Philips Electron Microscope in order to reveal internal structure. Oxygen isotopes have been analysed during two sessions. The first set of data was collected using the original mount where the grain was set in the resin attached to the glass slide. This resulted in the two complications: (i) standard zircon has to be analysed from the separate mount and (ii) the lunar zircon grain was rased in the holder compared to the standard. In order to investigate, if the elevated oxygen compositions observed during this session could have resulted from this difference in geometric configuration during the standard and sample analyses, the lunar zircon was extracted from the original mount, remounted with the standard chip in the new resin disk and reanalysed during the second session. All analyses made during the first session show delta O-18 values heavier than 6.0%. The second set of data has a wider spread of delta O-18 values with some values as low as 5.6%. Nevertheless, a half of observed delta O-18 values in this set is also higher than 6.0%. Slightly lighter oxygen compositions observed during the second session indicate possible dependence of measured delta O-18 values on the geometry of analysed samples. Presence of zircons with similar heavy oxygen isotope compositions on the Moon, which neither had liquid water or felic crust similar to that on the Earth nor ever developed regime similar to plate tectonics, suggests that other mechanisms can be responsible for elevated delta O-18 values in zircons. This implies that there is no support for the presence of an ocean on the surface of the early Earth and as the ocean appears to be an essential ingredient for the plate tectonics, there is no basis for belief that this mechanism was operating in the early history of the planet